Non-unitary Wightman CFTs and non-unitary vertex algebras

It is a fundamental mathematical challenge to establish a rigorous axiomatization of quantum field theory (QFT), and in general this problem remains wide open except in very specialized contexts. In recent years, axiomatic QFT has received particular attention in the context of two-dimensional chiral conformal field theories (CFTs), as these theories are sufficiently structured to enable a rigorous mathematical treatment while at the same time exhibiting a wide variety of mathematical connections (to operator algebras and subfactors, to representation theory and modular tensor categories, to vector-valued modular forms, and to many other areas). There are many proposed axiomatizations of two-dimensional chiral CFTs, each of which captures different aspects of the physical theory, and none of which have been rigorously demonstrated to provide a complete description of CFT. It is conjectured that these different axiomatizations are essentially equivalent, and there have been recent breakthroughs in comparing different axiomatizations under certain technical hypotheses [CKLW18, Ten19a].

In this article we demonstrate the equivalence of two well-known axiomatizations of two-dimensional chiral CFTs. We establish this equivalence in the most general way possible, without any reliance on auxiliary technical hypotheses or restrictions on the models under consideration, such as the existence of an invariant inner product (“unitarity”). Proving equivalences at this level of generality has largely been viewed as an aspirational (but not necessarily feasible) goal of axiomatic QFT, which we achieve here through a detailed analysis of the mathematical structures in question.

The first axiomatization that we consider is the non-unitary version of the (bosonic) Wightman axioms on the unit circle $S^{1}\subset{\mathbb{C}}$, with Möbius symmetry (i.e.​ symmetry group $\operatorname{M\ddot{o}b}:=\mathrm{PSU}(1,1)$, the holomorphic automorphisms of the unit disk). The key data of such a theory is a collection ${\mathcal{F}}$ of operator-valued distributions (or Wightman fields) acting on a common invariant vector space of states ${\mathcal{D}}$, along with a compatible positive-energy representation of $\operatorname{M\ddot{o}b}$.

The second axiomatization that we consider is (${\mathbb{N}}$-graded, bosonic) Möbius vertex algebras. These are vertex algebras graded by non-negative integer conformal dimensions, with symmetry given by the complexified Lie algebra $\mathfrak{su}(1,1)_{\mathbb{C}}\cong\mathfrak{sl}_{2}({\mathbb{C}})$.

We prove the following main result.

Our result does not require unitarity or the existence of an invariant bilinear form, and we do not require that the homogeneous subspaces for the grading by conformal dimensions be finite-dimensional. There are many important examples of CFTs arising in mathematical and theoretical physics which require this level of generality, and in particular non-unitarity arises from the CFT-approach to classical critical phenomena, and from string theory. Specific examples include the non-unitary Virasoro minimal models, affine vertex algebras at non-critical level (both universal and simple quotient), bosonic ${\mathbb{N}}$-graded affine W-algebras (again universal and simple quotient), and the $\beta\gamma$-ghost vertex algebra with central charge $c=2$ (along with other “$A$-graded” vertex algebras which arise in logarithmic conformal field theory). As a result of our theorem there are canonical Wightman CFTs associated to these models, which demonstrates significant functional analytic regularity that is not otherwise apparent.

The constructions going from Wightman CFTs to vertex algebras and back are given in Sections 3.1 and 3.2, respectively. These are shown to give an equivalence of categories in Section 3.3. The vertex algebra ${\mathcal{V}}$ associated to a Wightman CFT with domain ${\mathcal{D}}$ is constructed as a certain subspace ${\mathcal{V}}\subset{\mathcal{D}}$. Conversely, the Wightman CFT associated to ${\mathcal{V}}$ is constructed as an extension ${\mathcal{V}}\subset{\mathcal{D}}\subset\prod{\mathcal{V}}(n)$. We note that at this level of generality (allowing each weight space ${\mathcal{V}}(n)$ to be infinite-dimensional) it is not a priori clear that there is a single Wightman CFT for each vertex algebra, and it seems plausible that there could be families of ‘Wightman completions’ of a single vertex algebra. However, as a consequence of our result, there is indeed a unique Wightman CFT for each Möbius vertex algebra.

A very useful and inspiring heuristic discussion on the connection between Wightman CFTs and vertex algebras can be found in [Kac98, §1.2]. However, the arguments given there do not appear to be aimed to give precise mathematical details on this connection. More recently, three of the present authors gave a rigorous proof that unitary Möbius vertex algebras were equivalent to unitary Wightman CFTs possessing an additional analytic property called uniformly bounded order, provided that the homogeneous subspaces for the grading by conformal dimensions were finite-dimensional [RTT22]. The present article generalizes the previous result to possibly non-unitary theories, also dropping the requirements of uniformly bounded order and finite-dimensional weight spaces. As the techniques historically used to study Wightman theories involve careful analysis of the norm topology on the space of states, there is significant new work required to generalize our previous results to the non-unitary setting.

We also demonstrate in Section 4 that the correspondence constructed in this article is compatible with invariant bilinear forms, invariant sesquilinear forms, and invariant inner products.

We are left with a striking and clear correspondence between two well-known axiomatizations of two-dimensional chiral conformal field theory, without any reliance on additional technical hypotheses. We are motivated in part by the possibility to provide such an equivalence, which is not often possible in the wild landscape of axiomatic quantum field theory. We are also motivated by intriguing links between non-unitary conformal field theories and the unitary world of algebraic conformal field theory. Given a Wightman CFT on $S^{1}$ and an interval $I\subset S^{1}$, consider the algebra ${\mathcal{P}}(I)$ generated by Wightman fields $\varphi(f)$ smeared by test functions $f$ supported in the interval $I$. Such Wightman nets of algebras have been studied in the context of unitary quantum field theories [SW64], and there is a substantial effort underway to understand the relationship between unitary vertex algebras, unitary Wightman nets, and the usual nets of algebras of bounded observables (i.e.​ conformal nets) studied in algebraic conformal field theory [CKLW18, Ten19a]. On the other hand, as a result of our present work, there exists a Wightman net for every Möbius vertex algebra, including non-unitary ones. Such nets could give an avenue to apply methods generally used in the unitary framework of algebraic quantum field theory in the more general setting of non-unitary models. Previously such links have been probed only at the level of categories of representations [EG17]. As a first demonstration of the potential of this approach we prove a version of the Reeh-Schlieder Theorem (regarding the cyclic and separating property of the vacuum vector) for non-unitary theories in Appendix A.

Finally, there is strong motivation to understand functional analytic aspects of non-unitary vertex algebras as a part of studying links between algebraic and geometric aspects of the theory, as in [Hua99, Hua03]. More recently, analytic considerations of non-unitary vertex algebras have played a key role in the study of conformal blocks [Gui24a, Gui24b, GZ23], and such considerations also feature in the construction of functorial CFTs in the sense of Segal [Seg04].

In future work, it would be interesting to relate modules for vertex algebras to representations of the corresponding Wightman nets, which would fit into the broad program underway in the unitary setting to relate vertex algebra modules to representations in algebraic conformal field theory [Ten19b, Ten24, Gui21, Gui20, CWX]. Such relations should enable further correspondences between full two-dimensional conformal field theories in various approaches, cf.​ [Mor23, AGT23, AMT24].

An operator-valued distribution on the unit circle $S^{1}$ with domain a vector space $\mathcal{D}$ is a linear map

$$\varphi:C^{\infty}(S^{1})\to{\mathcal{L}}({\mathcal{D}}),$$

where ${\mathcal{L}}({\mathcal{D}})$ is the space of linear operators on ${\mathcal{D}}$. In this article, we will typically study operator-valued distributions whose domain ${\mathcal{D}}$ is infinite-dimensional, and we will require some topological considerations with respect to the action of sets of such distributions on ${\mathcal{D}}$.

If ${\mathcal{F}}$ is a set of operator-valued distributions on $S^{1}$ with a common domain ${\mathcal{D}}$, then a linear functional $\lambda:{\mathcal{D}}\to{\mathbb{C}}$ is called compatible with ${\mathcal{F}}$ if the multilinear maps $C^{\infty}(S^{1})^{k}\to{\mathbb{C}}$ given by

$$(f_{1},\ldots,f_{k})\mapsto\lambda\big{(}\varphi_{1}(f_{1})\cdots\varphi_{k}(f_{k})\Phi\big{)}$$

(2.1)

are continuous in the $f_{j}$ for all $\varphi_{1},\ldots,\varphi_{k}\in{\mathcal{F}}$ and $\Phi\in{\mathcal{D}}$. Note that multilinear forms $C^{\infty}(S^{1})^{k}\to{\mathbb{C}}$ are separately continuous if and only if they are jointly continuous since $C^{\infty}(S^{1})$ is a Fréchet space [Trè67, Cor. §34.2]. We write ${\mathcal{D}}_{\mathcal{F}}^{*}$ for the space of all linear functionals compatible with ${\mathcal{F}}$. Recall that a set ${\mathcal{X}}$ of linear functionals on ${\mathcal{D}}$ is said to separate points if for every non-zero $\Phi\in{\mathcal{D}}$ there is a $\lambda\in{\mathcal{X}}$ such that $\lambda(\Phi)\neq 0$.

If we imagine that ${\mathcal{F}}$ consists of a family of Wightman fields (i.e.​ the operators $\varphi(f)$ are smeared quantum fields), then it is natural that expectations $\lambda:{\mathcal{D}}\to{\mathbb{C}}$ should have the property that expressions (2.1) are continuous in the smearing functions $f_{j}$. Thus, the condition of regularity serves to exclude certain nonphysical actions that have the property that expectations cannot distinguish states. The following example illustrates the pathological behavior of nonregular actions.

Let $e_{n}\in C^{\infty}(S^{1})$ be the function $e_{n}(z)=z^{n}$. The condition that ${\mathcal{F}}$ acts regularly on ${\mathcal{D}}$ ensures that the operators $\varphi(f)$ are determined by the modes $\varphi(e_{n})$ in a certain sense that we will make precise below. This is in contrast with Example 2.2, in which $\tilde{\varphi}(e_{n})\tilde{\Omega}=0$ for all $n$ but $\tilde{\varphi}(f)\tilde{\Omega}\neq 0$ for some $f\in C^{\infty}(S^{1})$.

We now introduce certain topologies on ${\mathcal{D}}$ associated with the action of ${\mathcal{F}}$. We assume here that the reader is familiar with (or indifferent to) the fundamentals of topological and locally convex vector spaces, and defer the relevant background and additional details to Appendix B.

For a topological vector space $X$, a map $T:X\to{\mathcal{D}}$ is ${\mathcal{F}}$-weakly continuous precisely when $\lambda\circ T$ is continuous for all $\lambda\in{\mathcal{D}}_{\mathcal{F}}^{*}$. The family ${\mathcal{F}}$ acts regularly precisely when the ${\mathcal{F}}$-weak topology is Hausdorff. We will see in Lemma 2.8 below that for $\varphi\in{\mathcal{F}}$ the expressions $\varphi(f)\Phi$ are continuous in $f\in C^{\infty}(S^{1})$ when ${\mathcal{D}}$ is given the ${\mathcal{F}}$-weak topology, so indeed $\varphi(f)$ is determined by the modes $\varphi(e_{n})$ when ${\mathcal{F}}$ acts regularly.

There is a second natural topology on ${\mathcal{D}}$ associated with the action of ${\mathcal{F}}$. For $\varphi_{1},\ldots,\varphi_{k}\in{\mathcal{F}}$ and $\Phi\in{\mathcal{D}}$, let

$$S_{\varphi_{1},\ldots,\varphi_{k},\Phi}:C^{\infty}(S^{1})^{\otimes k}\to{\mathcal{D}}$$

be the linear map

$$S_{\varphi_{1},\ldots,\varphi_{k},\Phi}(f_{1}\otimes\cdots\otimes f_{k})=\varphi_{1}(f_{1})\cdots\varphi_{k}(f_{k})\Phi.$$

We equip the algebraic tensor product $C^{\infty}(S^{1})^{\otimes k}$ with the projective topology, for which continuous linear maps $C^{\infty}(S^{1})^{\otimes k}\to X$ correspond to continuous multilinear maps (see Appendix B).

Equivalently, the ${\mathcal{F}}$-strong topology is the finest locally convex topology on ${\mathcal{D}}$ such that expressions $\varphi_{1}(f_{1})\cdots\varphi_{k}(f_{k})\Phi$ are continuous in the functions $f_{j}$ (jointly, or equivalently separately by [Trè67, Cor. §34.2]).

We now have the following alternate characterizations of the regularity of an action of ${\mathcal{F}}$ on ${\mathcal{D}}$.

Both the ${\mathcal{F}}$-strong and ${\mathcal{F}}$-weak topologies are quite natural, and so it is not surprising that the fields ${\mathcal{F}}$ act continuously when ${\mathcal{D}}$ is given one of these topologies.

We now assume our vector space ${\mathcal{D}}$ is equipped with a family ${\mathcal{F}}$ of operator-valued distributions on $S^{1}$ as well as a representation $U:\operatorname{M\ddot{o}b}\to\operatorname{End}({\mathcal{D}})$, where $\operatorname{M\ddot{o}b}=\mathrm{PSU}(1,1)$ is the group of holomorphic automorphisms of the closed unit disk. As in [CKLW18, §6], for $\gamma\in\operatorname{M\ddot{o}b}$ we denote by $X_{\gamma}\in C^{\infty}(S^{1})$ the function

$$X_{\gamma}({\mathrm{e}}^{{\mathrm{i}}\vartheta})=-{\mathrm{i}}\frac{d}{d\vartheta}\log(\gamma({\mathrm{e}}^{{\mathrm{i}}\vartheta})),$$

which takes positive real values since $\gamma$ is an orientation-preserving diffeomorphism of $S^{1}$. For $f\in C^{\infty}(S^{1})$ and $d\in{\mathbb{Z}}_{\geq 0}$ we denote by $\beta_{d}(\gamma)f\in C^{\infty}(S^{1})$ the function

$$(\beta_{d}(\gamma)f)(z)=(X_{\gamma}(\gamma^{-1}(z)))^{d-1}f(\gamma^{-1}(z)).$$

(2.3)

An operator-valued distribution with domain ${\mathcal{D}}$ is called Möbius-covariant with conformal dimension $d$ under the representation $U$ if for every $\gamma\in\operatorname{M\ddot{o}b}$ and every $f\in C^{\infty}(S^{1})$ we have

$$U(\gamma)\varphi(f)U(\gamma)^{-1}=\varphi(\beta_{d}(\gamma)f)$$

as endomorphisms of ${\mathcal{D}}$. We say that a vector $\Phi\in{\mathcal{D}}$ has conformal dimension $d\in{\mathbb{Z}}$ if $U(R_{\vartheta})\Phi={\mathrm{e}}^{{\mathrm{i}}d\vartheta}\Phi$ for all rotations $R_{\vartheta}\in\operatorname{M\ddot{o}b}$.

We now present the not-necessarily-unitary version of the Wightman axioms for two-dimensional chiral conformal field theories on the circle $S^{1}$. Historically, the Wightman axioms have been closely entwined with unitary theories, where the space of states ${\mathcal{D}}$ possess an appropriate inner product. Non-unitary versions of the Wightman axioms have also appeared in various contexts such as the mathematical description of gauge fields, see e.g.​ [Str93, §6.4]. In this article we will generally refer to the non-unitary theories in question simply as Wightman conformal field theories for the sake of brevity.

A Möbius-covariant Wightman CFT is a quadruple $({\mathcal{F}},{\mathcal{D}},U,\Omega)$, but we will frequently refer to the family ${\mathcal{F}}$ of fields or the domain ${\mathcal{D}}$ as a (Möbius-covariant) Wightman CFT when the remaining data is clear from context.

Let $e_{j}(z)=z^{j}\in C^{\infty}(S^{1})$, and let

$${\mathcal{V}}(n)=\operatorname{span}\{\varphi_{1}(e_{j_{1}})\cdots\varphi_{k}(e_{j_{k}})\Omega\,|\,j_{1}+\cdots+j_{k}=-n\}.$$

By Möbius covariance (W1) the vectors in ${\mathcal{V}}$ have conformal dimension $n$, or in other words $U(R_{\vartheta})$ acts on ${\mathcal{V}}(n)$ as multiplication by ${\mathrm{e}}^{{\mathrm{i}}n\vartheta}$.

Let ${\mathcal{D}}_{\mathcal{F}}^{*}\cap{\mathcal{V}}(n)^{*}$ denote the space of linear functionals $\lambda\in{\mathcal{D}}_{\mathcal{F}}^{*}$ such that $\lambda|_{{\mathcal{V}}(m)}=0$ when $m\neq n$. The following technical observations will be essential in constructing vertex algebras from Wightman CFTs.

We will later see that for every $\lambda\in{\mathcal{V}}(n)^{*}$ there is a unique extension to a linear functional in ${\mathcal{D}}_{\mathcal{F}}^{*}$ that vanishes on ${\mathcal{V}}(m)$ (when $m\neq n$) – see Proposition 3.14.

The notion of Wightman CFT presented in Definition 2.9 generalizes the unitary notion of Wightman CFT considered in [RTT22] in several ways. Most notably, the domain ${\mathcal{D}}$ does not have an inner product. We also do not assume that $\Omega$ is the unique $\operatorname{M\ddot{o}b}$-invariant vector up to scale, and we do not require that the eigenspaces for the generator of rotation are finite-dimensional.

The requirement that ${\mathcal{F}}$ act regularly on ${\mathcal{D}}$ is necessary, as there exist pathological examples of quadruples $({\mathcal{F}},{\mathcal{D}},U,\Omega)$ that satisfy all of the requirements to be a Wightman CFT except for the regularity of the action of ${\mathcal{F}}$. Indeed, we can refine Example 2.2 to produce a quadruple with non-regular action for which the only finite-energy vectors are scalar multiples of the vacuum, despite ${\mathcal{D}}$ being infinite-dimensional.

With these distinctions in mind, we give the corresponding notion of vertex algebra after some brief preliminaries. Let $\operatorname{Lie}(\operatorname{M\ddot{o}b})$ be the three-dimensional real Lie algebra of $\operatorname{M\ddot{o}b}=\mathrm{PSU}(1,1)$. If $\operatorname{M\ddot{o}b}$ is regarded as a subgroup of the group $\mathrm{Diff}(S^{1})$ of orientation-preserving diffeomorphisms of the unit circle $S^{1}$, then $\operatorname{Lie}(\operatorname{M\ddot{o}b})$ is identified with a three-dimensional subspace of the space of smooth vector fields $\mathrm{Vect}(S^{1})$ on $S^{1}$. Each vector field is identified with a differential operator $f({\mathrm{e}}^{{\mathrm{i}}\vartheta})\frac{d}{d\vartheta}$ for some smooth function $f({\mathrm{e}}^{{\mathrm{i}}\vartheta})$, and the Lie bracket is given by $[f\frac{d}{d\vartheta},g\frac{d}{d\vartheta}]=(f^{\prime}g-fg^{\prime})\frac{d}{d\vartheta}$, where $f^{\prime}$ denotes $\tfrac{df}{d\vartheta}$. Note that this bracket is the opposite of the bracket of vector fields, which is the natural choice when identifying $\mathrm{Vect}(S^{1})$ with the Lie algebra of $\mathrm{Diff}(S^{1})$. The complexification $\operatorname{Lie}(\operatorname{M\ddot{o}b})_{\mathbb{C}}\cong\mathfrak{sl}(2,{\mathbb{C}})$ of $\operatorname{Lie}(\operatorname{M\ddot{o}b})$ is spanned by the elements $\{L_{-1},L_{0},L_{1}\}$, where $L_{m}$ is the complexified vector field $-{\mathrm{i}}{\mathrm{e}}^{{\mathrm{i}}m\vartheta}\frac{d}{d\vartheta}$. The vector fields $L_{m}$ satisfy the commutation relations

$\displaystyle[L_{m},L_{n}]=(m-n)L_{m+n},\qquad m,n=-1,0,1.$

In a representation of $\operatorname{Lie}(\operatorname{M\ddot{o}b})_{\mathbb{C}}$, we will frequently abuse notation and write $L_{k}$ for the operator corresponding to the vector field indicated above.

If ${\mathcal{V}}$ is a vector space we write $\operatorname{End}({\mathcal{V}})[[z^{\pm 1}]]$ for the vector space of formal power series in $z^{\pm 1}$ with coefficients in $\operatorname{End}({\mathcal{V}})$. Given $v\in{\mathcal{V}}$ and $A(z)=\sum_{n\in{\mathbb{Z}}}A_{n}z^{n}\in\operatorname{End}({\mathcal{V}})[[z^{\pm 1}]]$, we have a formal series $A(z)v=\sum_{n}A_{n}vz^{n}$ with coefficients in ${\mathcal{V}}$. For any $B\in\operatorname{End}({\mathcal{V}})$ we have $[A(z),B]=\sum_{n}[A_{n},B]z^{n}\in\operatorname{End}({\mathcal{V}})[[z^{\pm 1}]]$. If $B(w)$ is another formal series in a second formal variable $w$, then the expression $[A(z),B(w)]$ makes sense as a formal series in $z^{\pm 1}$ and $w^{\pm 1}$.

We can now precisely specify the flavor of vertex algebras that we will consider.¹¹1The term Möbius vertex algebra has been used in the literature to describe various slightly different notions (see e.g.​ [BK08, HLZ14, Hua20, Kac98]). In some cases, authors include the possibility of fermionic fields, with the corresponding super version of the locality axiom; the term Möbius vertex superalgebra is also used in this case. Additionally, some authors replace our ${\mathbb{N}}$-grading with a more general grading. We do not foresee any significant obstacles to generalizing our results to Möbius vertex superalgebras graded by a lower-bounded subset of $\tfrac{1}{2}{\mathbb{Z}}$.

For $v\in{\mathcal{V}}$, we write $Y(v,z)=\sum_{m\in{\mathbb{Z}}}v_{(m)}z^{-m-1}$, where $v_{(m)}\in\operatorname{End}({\mathcal{V}})$ are called the modes of $v$. A vector $v\in{\mathcal{V}}$ is called homogeneous (with conformal dimension $d$) if it lies in ${\mathcal{V}}(d)$. As a consequence of the $L_{0}$-commutation relation, when $v$ is homogeneous with conformal dimension $d$ we have $[L_{0},v_{(m)}]=(d-m-1)v_{(m)}$. Hence $v_{(m)}$ maps ${\mathcal{V}}(n)$ into ${\mathcal{V}}(n+d-m-1)$. A vector $v\in{\mathcal{V}}$ is called quasiprimary if it is homogeneous and $L_{1}v=0$.

We record two useful identities satisfied by the modes of a vertex operator (see [Kac98, §4.8]), the Borcherds product formula:

$$\big{(}u_{(n)}v\big{)}_{(k)}=\sum_{j=0}^{\infty}(-1)^{j}\binom{n}{j}\left(u_{(n-j)}v_{(k+j)}-(-1)^{n}v_{(n+k-j)}u_{(j)}\right),$$

(2.5)

and the Borcherds commutator formula:

$$[u_{(m)},v_{(k)}]=\sum_{j=0}^{\infty}\binom{m}{j}\big{(}u_{(j)}v\big{)}_{(m+k-j)}.$$

(2.6)

Note that when the sums of operators on the right-hand sides of (2.5) and (2.6) are applied to a vector, all but finitely many terms vanish.